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A337672 Numbers with binary expansion Sum_{k = 0..w} b_k * 2^k such that the polynomial Sum_{k = 0..w} (X+k)^2 * (-1)^b_k is constant. 0
0, 9, 150, 153, 165, 195, 2268, 2282, 2289, 2364, 2394, 2406, 2409, 2454, 2457, 2469, 2499, 2618, 2646, 2649, 2661, 2702, 2709, 2723, 2829, 2835, 3126, 3129, 3150, 3157, 3171, 3213, 3219, 3339, 3591, 34680, 34740, 34764, 34770, 34785, 35576, 35700, 35756 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Leading 0's in binary expansions are ignored.

Positive terms are digitally balanced (A031443).

If m belongs to the sequence, then A056539(m) also belongs to the sequence.

If m and n belong to the sequence, then their binary concatenation also belongs to the sequence (assuming the concatenation with 0 is neutral).

LINKS

Table of n, a(n) for n=1..43.

Index entries for sequences related to binary expansion of n

EXAMPLE

The first 16 integers, alongside their binary representations and associate polynomials, are:

  k   bin(k)  P(k)

  --  ------  --------------

   0       0               0

   1       1            -X^2

   2      10           2*X+1

   3      11    -2*X^2-2*X-1

   4     100       X^2+6*X+5

   5     101      -X^2-2*X-3

   6     110      -X^2+2*X+3

   7     111    -3*X^2-6*X-5

   8    1000   2*X^2+12*X+14

   9    1001              -4

  10    1010           4*X+6

  11    1011   -2*X^2-8*X-12

  12    1100          8*X+12

  13    1101    -2*X^2-4*X-6

  14    1110        -2*X^2+4

  15    1111  -4*X^2-12*X-14

We have constant polynomials for k = 0 and k = 9, so a(1) = 0 and a(2) = 9.

PROG

(PARI) is(n) = { my (b=Vecrev(binary(n))); poldegree(p=sum(k=1, #b, ('X+k-1)^2 * (-1)^b[k]))<=0 }

CROSSREFS

Cf. A031443, A056539, A133468.

Sequence in context: A183439 A296017 A218350 * A027018 A326012 A089916

Adjacent sequences:  A337669 A337670 A337671 * A337673 A337674 A337675

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Sep 15 2020

STATUS

approved

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Last modified November 30 15:30 EST 2021. Contains 349420 sequences. (Running on oeis4.)